THB BCONOMIC RESBARCH INSTITUTB

Memorandum No.16

Correcting for Seasonality: I The Regression

Method, with an Application to Irish Data

R.C. Geary

Let the data be observed for T consecutive

years with k(= 12 for months~ = 4 for quarters) length

of seasons. The observations are Yt’ t = kt~ + s,

tT = 1,2,...,T, s = 1,2,...,k. Set up the regressiont’

(i) Yt=blXtl +b2xt2 +o..+bpXtp+clztl +c2zt2 +...+ckztk +ut,

where Yt "-: Yt - ~" The xti are the trend (regarded as

orthogonal) and the ztj are the seasonality variables,

ut the random residual. The ztj elements are all zeros

or u nities~ Their Tk x k matrix Z is

I

where I is the unit k x k matrix and these are T such

matrices. Since the xti are orthogonal,

zt xti Xti’ (i~-i’) = O o

Similarly for the ztj.

(4) ~’t ztj ztj’(j~j’) = 0

The regression solution of (i) in the b andi

c, is found fromJ

(i) %1t XtiYt = biZxt2i + X.C EtXtiZt J j’

(5)(ii)

Et ztjYt = Z.b Z± i tztjxti + T cj, j = 1,8, ..~,k

On account of the or~hogonal property this

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system is very easily set up and solved.

be shown from (i) that

(6) k~, e. =0,

j=l 1

It can readilF

which is a desirable property in the seasonality

components and is a check on the solution of (5).

To solve (5) the only sum-product computations

required are

S:t = ZtxtiYt = Z~xtiYt, i = 1,2, .... , p

All the other terms in (5) are summations the ztj are

zeros or unities. The method of solution involves

substitution for the c. from (5) (ii) into (5) (i),]

solving for the b. and substituting back into (5) (ii)i

.. The method will be illustrated byto obtain the c

application to the data in Table i.

Application to Raw Data

Table i. Quarterly Output of Blectricity in

Five Years 1959-1963

¯ ~WHm,

Quarter

Year I II III IV To t al

1959 572 437 417 593 2,019

1960 646 470 464 658 2,238

1961 668 507 491 698 2,364

1962 754 565 538 756 2,611

1963 852 617 578 813 2,860

Su m 3~492 2,594 2,488 3, 518 I~,092Z1 Z2 Z3 Z4

L.

Source: ITJSB.

It is proposed to use the first 2(=p) orthogonal

polynomials for trend. Also k = 4, T = ~5, kT = 20,

so that Y = 12,092/20 = 604.6, We also require the

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orthogonal polynomial elements shown in Table 2 set

out in 5 sets of 4.

Table 2, Orthogonal Polynomial Blements.

Set Xtl xt2

1-1959 -19 -17 -15 -13 57 39 23 9

2-1960 -ii - 9 - 7 - 5 - 3 -13 -9.1 -27

3-1961 - 3 - 1 1 3 -31 -33 -33 -31

4-1962 5 7 9 ii -27 -91 -13 - 3

5-1963 13 15 17 19 9 25 39 57

Sum -15 - 5 5 15 5 - 5 - 5 5

Source: Statistical Tables by R.A. Fisher andF, Yates, Fifth Ed. page 91 (n’ = 20)

The two sum-products are

ZtxtlYt = 16,412; Ztxt2Yt = 3,036.

The equation system (5) then is

C1

16,412 = 2,660bI -15cI - 5c2 + 5c(i)

3 + 15c4

3 + 5c43?036 = 17,556b2 + 5cI - 5c2 - 5c...................... -~ F J

C~

(The coefficients of bI and b2 are tabled in the source)

(ii)

3,492 - 5 x 604.6 = 469 = -15bI + 5b2 + 5c1

2)594 - " =-429 = - 5bI - 5b2 + 5c2

2,488 - " =-535 = 5bI - 5b2 + 5c3

5,518 - " = 495 = 15bI + 5b2 + 5c40

(Note, by addition of (ii), ~. = O)3

From (ii), noting definition of C1 and C2,

C1 = --28 - lO’.Ob1

C2 = 1,928 + 20b2,

Substituting in (i) ,

16,412 = 2,660 bI -28-i00bI or bI=16,440/2,560=6,421875

3,036 = 17,556 b2+1,928+ 20b2 or b2= i,i08/17, 576 =0 .06 Z041

Finally, on substituting these values for bI and b2 in

(ii) we have

5cI = 565.019; 5c2 =-396.575; 5c

\

with check that the sum is zero.

3 = -566.794; 5c4 = 398.357

For computational

purposes it will be convenient to set down the values

of c + Y. These are3

cI + Y = 717,6

c + Y = 525dZ9

C~ + Y = 491.2

c4 + Y = 684. $

Having determined the coefficients bl, b2, Cl, c2, c3~ c4,

the value of R9" can be found~ This is ,9800 which might

be regarded as satisfactorily large. But is it the best

we can do?

Appliqation to Logarithmic Data

The answer is no. The implication of this

arithmetical approach is that the actual production in

any quarter in any year is to be found by adding the

correction (i.e. the appropriate cj) to the tre~d,; this

correction is the same for all years. This assumption

is somewhat unreal in a rapidly increasing phase of

expansion. As a rough test from Table i it will be

observed that the ranges (all between quarters III

and IV) in the successive five years were 176, 194,

207, 218~ 235, or 8,7, 8.7, 8.8, 8.3, 8.2 of annual

production. While there may be some slight tendency

towards a declining amplitude there can be no doubt

that the hypothesis of seasonality acting proportionately

should yield more satisfactory results with this data

than to assume additivity.

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We accordingly repeat the foregoing

calculations but now using log Yt instead of Yt" The

results are as follows (using primed letters for

corresponding symbols used in the foregoing arithmetical

cI

c2

c3

c6

Jcase).

Y’ = 2.7726

bI’ = .024609375

b2t = .0433910

’ + Y’ = 2.8536

+ Y’ = 2.7162

+ Y’ = 2.6896

w’ + Y’ = 2.8310

The regression yields the satisfactorily high value of

.9913 for R2. There can be little doubt that the

logarithmic approach is to be preferred. The results

are compared in Table 3. The "calculated" values are

those using formula (I) with ut = O. For the

logarithmic regression the antilogs are, of course~

shown. These resulted in a slight discrepancy in the

five-year output of 12,099 in givin~ a figure of 12,088:

the 4 units were distributed 8o as to give the correct

total.

The actual series (Table 3, Column 2) and the

two calculated series (Columns 3 and 4) are graphed on

the appended Diagram.

s Table 3.

J

Actual and Calculated Values of Output of Electricity,

Quarterly 1959-1963

KWHm.

CalculatedDeviation (absolutevalue), frgm ,actual

Qu art er ActualDifference,

cols. 5-6(i) Original (ii) Logar- (i) Original (ii) Logar-

data ithmic data ithmic

1 2 3 4 5 6 7

1959 I 572 599 587 27 15 +12I! 437 419 436 18 I +17III 417 396 418 21 i +20IV 593 501 592 8 i + 7

1960 I 646 647 636 1 i0 - 9II 470 466 472 4 2 + 2Ill 464 445 454 19 i0 + 9IV 658 651 641 7 17 -i0

1961 i 668 696 690 28 22 + 6II 507 517 514 i0 7 + 3III 491 495 493 4 2 + 2IV 698 702 698 0 +

1962 I 754 748 751 6 3 + 3IX 563 569 559 6 4 + 2IIi 538 548 538 i0 0 +i0IV 755 755 761 /.- 1 5 - 4

196 3 I 852 802 820 50 32 +i 8II 617 623 611 6 6 +0III 578 603 588 25 i0 +i 5IV 813 810 833 3 i0 -17

To t al 12,092 12,092 12,092 258 168 9O

!

!

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Testing for Relative Efficiency

Are the deviations in columns 5 and 6

significantly different? The two entries for each

of the twenty units may be regarded as "treatments".

The nul-hypothesis is that these are not significantly

different. It will be noted from column 7 that in

only 4 cases out of 20 are the column 5 deviations

less than those in column 6. On the nul-hypothesis

the number of - or + signs would have a probability

distribution of

+

when the probability of 4 or fewer - or + signs being

found is ~Oll8, so small that the nul-hypothesis is

rejected.

Procedure

The seasonality factors Cl, c2, c3, c4

based on the original data or some transform of these

(e.g. the logarithm) would be calculated for the

latest five calendar years and applied to correct the

four quarters of the following year. As soon as the

data for the four quarters of the current year become

available the same procedure is adopted. The actual

data displayed in the example worked out above will

be used for different series and different sets of

years. Only the following data are specific to

each series: Zl, Z2~ Z3~ Z4 (see Table i)~

Sl = Et xtl Yt’ $2 = Ext2 Yt" The actual formulae

for the c could be written down as linear expressionsJ

in these symbols and the "constant" symbols. The

solution in arithmetical terms is so simple that there

is not much point in doing so.

As to the actual procedure as applied to

the data for 1964 we have

log seasonality Antilog Actualfactors 1959-63

(i) (2) (3) (4)

cI’ = 0:0810 ioe050 885D

c2 ~ = T, 94~6 0. 8782 676

c3T = T. 9170 0.8260

c4’ = 0.0584 1.1440

OutputCorrected

(s)734

770

The product of the factors in column (3) is unity°

Between the first and second quarter of 1964 seasonally

corrected output increased by 86 KWHm. or by 4.9%.

Short term Forecasting

The logarithmic regression fit is so

remarkably good that confidence might be reposed in

using formula (i) (without ut) for forecasting the

quarterly output in the short term from I 1954.

this the formulae for Xtl and xt2 are required°

are as follow~’-

FO r

Th ey

Xtl ~ 2t + 1

t2 + t - 33xt2 =_

The serial number for I 1964 is i0~

for these polynomi~l terms and the ztj

shown in Table 4.

The actual values

elements are

Table 4.

I 1964 Ii 1964 III 1964 IV 1964 I 1965 Co ef fi ci ent s

i0 ii 12 13 14(log arit hmi c )

Xtl 21 23 25 27 29 0.024609375

xt2 77 99 123 149 177 -O.04~3910

Ztl1 0 0 0 1 2.8536~

Zt2 0 1 0 0 0 2.7162~

zt3 0 0 1 O O 2.6896~

zt4 0 0 0 1 O 2. 8310~

Calculated

Log Yt 2. 9478 2 o8189 2. 8007 2.9504 2.981Z

Y 887 659 892t

632 95 8 KWHm.

Actual Y 885 676 T!

t

l

!

* Including mean log Y = 2.7726

-I0-

The entries at log Yt are the sum products of the

last co!u:~n by the other columns. The correspondence

is excellent in i 1964, actually better than is to

be expected normally, as reference to column 6 of

Table 3 will show. The mean deviation for that colun~j

is 8°4 and the standard deviation 9.’7 so that the

deviation of 17 KWHm for II !964 while somewhat large~:’

than might usually be found is not yet to be regarded

as a change in the 1959-196~ trend.

Comparison of Seasonality Correction Divisors

The seasonality correction divisors are co~n ....

puted by three methods, the two methods described

above and a third based on the adjusted movin~ ~V~

method. The results are Ghown in Table 5.

Table 50 Seasonality Correction Divisors Com.-puted by Three Methods, for Quarterly~!ectricity Output 1959-1963

Quarter

I

II

IIi

IV

Regression

Orlglnaldata

1. o.08

0.878

0.8~5

1.143

Log orig.data

io 205

0.878

0.826

I. 144

Moving

average

I. 208

0.880

Oo 87’:

i o 140

In each case the four divisors have been adjusted so

that their product is unity.

The results are almost identical° For the

application of the moving average method (last column)

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the actual moving average results are corrected for

the well-known aberrations at high and low points on

the moving average curve, to determine trend, using

a method described in Memorandum No. 17 : estimates

of divisors for each quarter are the five" year geometric

means of the quotient of original data by trend.

Concluding Remarks

It is not to be expected that the method

described here will yield quite so satisfactory results

when used with other major series: for example, in

the Third French Plan the percentage increases (to the

unit place) forecast were identical with the actual

increases in each case in electricity, gas and petro-

¯ leum production . Quite a large amount of experi-

. mental work has been completed in the Institute on

seasonality correction using the moving average

method corrected for change in trend - to be described

in another BRI Memorandum - and this work shows that

(1) major irish time data fluctuate

seasonally with large amplitude;

at the quarterly level these fluctuatie~o

exhibit marked regularity;

(3) accordingly, any good method for season-

ality correction, applied at the quarterly

level is likely to yield reliable

results.

Lloyds Bank Review~ June, 1964~ p.24

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The prospects of being able to appraise the trend of

the Irish economy at quarterly intervals are distinctly

promising.

August 1964.

8OO

II III IV1959

I II III IV I II III IV 1 II III IV

1961 I~2

P

:G+.<